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Jones, Member, IEEE Abstract This paper proposes a novel approach in the design, construction and analysis of a continuum robot.

2 Proceedings of the 007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 9 - Nov, 007 WeA.4 Design, Construction, and Analysis of a Continuum Robot Srinivas Neppalli, Student member, IEEE, Bryan A. Jones, Member, IEEE Abstract This paper proposes a novel approach in the design, construction and analysis of a continuum robot. The paper examines the drawbacks of two existing designs and proposes a new mechanical design that uses a single latex rubber tube as the central member, providing a design that is both simple and robust. Next, a new, simplified method of modeling kinematics is introduced. A novel verification procedure is then applied to examine the validity of the proposed model in two different domains of applicability and could be used to verify many other models that are constructed based on similar assumptions. Finally, a two-level electrical control scheme enables rapid prototyping. Index terms Biologically inspired robot, continuum robot, kinematics, trunk. I. INTRODUCTION Continuum robots are inspired by the biological world. These robots feature a backbone-less structure similar to such biological counterparts as snakes, elephant trunks and octopus arms. Like these organisms, a continuum robot can use the entire length of its arm to grasp objects of different shapes and sizes. It wraps its body around the object until it has a firm grip on the object, similar to an elephant using its trunk to move logs. Potential applications of continuum robots include navigation through congested and unpredictable environments where a continuum robot can be used for underground or underwater exploration. A continuum robot with its unique abilities can reach places that are usually inaccessible for rigid link robots and hostile for human beings. In case of an accident, the loss will be minimal because the body of the continuum robot is made of a latex rubber tube. The costly control circuitry and motors are always on the top of the ground. Underwater operation is also possible because of the absence of any electrical components throughout the body of the continuum robot. These robots are also known for their wide range of grasping abilities. They can firmly grasp objects of different shapes and sizes as shown in []. The grasping can also be performed by various biologically inspired methods; for example, the continuum robot featured in [] can grasp a plastic box like an octopus with the help of the suction cups attached to its body. It is to be noted that the robot featured Manuscript received April 7, 007. Srinivas Neppalli is with the Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 3976 USA (phone : ; sn40@ece.msstate.edu). Bryan A. Jones is with the Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 3976 USA (phone: ; fax: ; bjones AT AT ece.msstate.edu).. in [] is different from the one proposed in this paper but both the robots share a common basic principle. However, in spite of the large number of prototype and commercial continuum robots constructed [], several challenges remain in the design and analysis of continuum robots. First, the development of a robust continuum robot that is easy to build and control as well as portable and versatile remains an open research area. The design presented in this paper and shown in Fig. begins to address these concerns. Second, the derivation of the kinematics for these manipulators lacks a straightforward, simple formulation which this paper provides. Finally, little verification of the modeling assumptions has been performed; experimental results presented in this paper demonstrate the regions in which the model applies, validating these assumptions. This paper is organized as follows. After introducing continuum robots we present a novel approach in design and construction of continuum robot in Section II. Section III describes a simplified method of modeling kinematics and introduces a novel verification procedure to examine the validity of the proposed model. In Section IV we explain the proposed electrical design to control a continuum robot. The paper concludes in Section V. II. DESIGN AND CONSTRUCTION OF A CONTINUUM ROBOT This paper contributes a novel design combining the simplicities of Air-Octor [3] with the agility of OctArm [4], resulting in a continuum robot that is not only mechanically simple and easy to build but also robust and efficient. This Nylon sleeve Latex rubber tube To air-inlet Cable guides Hose Clamps Sealed end cap Cables Fig. Continuum robot constructed based on the proposed design. A latex rubber tube is used as the central member that is surrounded by three cables mutually separated by 0. See Fig. 3 for a cross-sectional view of the trunk /07/$ IEEE. 503

Cable Y Nylon Sleeve r = φ X 0 o Rubber S Cable Cable 3 Fig. Manipulator variables s,, and φ, where φ gives the rotation in the xy plane, defines the curvature and s gives the length of the trunk.

3 Cable Y Nylon Sleeve r = φ X 0 o Rubber S Cable Cable 3 Fig. Manipulator variables s,, and φ, where φ gives the rotation in the xy plane, defines the curvature and s gives the length of the trunk. paper examines Air-Octor and OctArm, where Air-Octor is a simpler design to construct and the OctArm offers better performance in grasping and whole arm manipulation than the former. OctArm is flexible, elastic and has good strength, but is complex to build and control because of the multiple pressurized central members that make the design mechanically challenging. Air-Octor, on the other hand, is much less complex to build and control because of the single central member and the use of cables as actuators but lacks flexibility and strength due to high cable friction which cannot be overcome by low pressure in the central member, resulting in cable binding which in turn causes undesirable movements of the trunk. The trunk presented in this paper is not only easy to build and control but also provides good strength and flexibility for the continuum robot. This paper presents a novel approach for building a continuum robot that replaces the dryer hose, the problematic central member of Air-OCTOR, with a latex rubber tube that has more strength and flexibility [5]. Like many previous designs [], the central member is surrounded by three cables separated by 0 degree intervals []. Fig. 3 shows a cross-sectional view of the trunk explaining the arrangement of cables around the trunk. No. TABLE I VARIOUS COMBINATIONS OF TUBES AND SLEEVES Outer Diameter in Thickness in mm Nylon Sleeve Size in Type of cable guides used Dual Layer Nylon Sleeve Cable ties Cable Ties Fig. 3 Cross-sectional view of the trunk showing three cables that are mutually separated by 0. Cable ties run through the nylon sleeve to form small loops through which the cables can be passed freely. See Fig. for a picture of the actual trunk. The lengths of these three cables define the shape of the continuum robot [6]. The central member is made up of a latex rubber tube covered with an expandable nylon sleeve. A rubber tube is a better choice for building a continuum robot than a dryer hose (used in Air-OCTOR) because of its flexibility, elasticity and strength. A rubber tube can handle pressures up to 483 kpa whereas a dryer hose can be pressurized only up to 3.8 kpa [5]. In addition this approach uses only one pressurized member per section which makes it a simpler mechanical design than that of OctArm. The length of this member can be changed by varying the pressure in the member. When pressurized, a rubber tube expands in all directions like a balloon. To restrict the expansion longitudinally without losing its cylindrical shape, it is covered tightly with an expandable nylon sleeve. Various sizes of rubber tubes and matching sizes of nylon sleeves that were experimentally determined are shown in Table I. The rubber tube is sealed on both sides with a metal tube fitting. One end is perma- Pressure in kpa TABLE EXPANSION AT VAROIUS PRESSURES Length in Length in 3 Length in Dual Layer Nylon Sleeve

4 Z Arc of trunk extending along + Z axis c φ r = nently blocked. A small air inlet is placed on the other end. Hose clamps are used to hold the sleeve, tube and fittings in place.the physical dimensions of the tube and sleeve affect the amount of expansion at a given pressure. The results after experimental verification with different combinations of tubes and sleeves and their expansions at various pressures are tabulated as shown in Table. is the best combination among those verified, demonstrating an extension of 34% at 483 kpa. Because the central member of Air-Octor can withstand only a very low pressure (3.8 kpa) the cable guides used offer a considerable amount of friction compared to the pressure, resulting in binding of cables. In addition to increasing the pressure in this new prototype, two methods were examined for the use of lower-friction cable guides to avoid binding. In the first method cable ties are used as cable guides. Cable ties hold the cables to the sleeve that covers the trunk and run through the sleeve and form small loops through which the cables can be passed freely. A hose clamp is used to hold the cables on the terminating side. In the second method, two layers of nylon sleeve are used as cable guides. The inner layer covers the rubber tube tightly and the outer layer holds the cables running through it. A nylon sleeve offers low friction comparable to cable ties but the outer layer of the sleeve restricts the expansion of rubber tube. Therefore we choose the first method as the cable guiding mechanism. Several experiments were conducted using various tubes and types of cable guides. Their expansions at different pressures are shown in Tables and. III. MODELING AND VERIFICATION OF A CONTINUUM ROBOT Though the circular arc assumption made by the model proposed in section III and shared by much of the continuum kinematics literature [6-0] has been widely used, this underlying assumption has not been experimentally verified. Y Circle center at r cosφ r sinφ 0 Fig. 4 Simplified model of kinematics that can be derived through purely geometrical means. Arc of the trunk extends along the +z axis and bends along the direction φ in the xy plane. X This paper describes a novel procedure to experimentally verify this assumption for a continuum robot for two different cases (with and without gravity). A. Modeling An analysis of the dynamics of a planar flexible beam undisturbed by external forces and subject to a torque applied to the end of the beam shows that the beam forms a curve of constant curvature, which is an arc of a circle []. This constant-curvature assumption provides a basis for much of the existing kinematic analysis of continuum robots [6-0]. The following paragraphs present a novel, concise derivation of the kinematic results of this assumption, followed by an experimental examination of the validity of this assumption. To determine the kinematics of an arc, note that the motion due to the trunk is a classical rigid motion: a revolute joint placed at the center c of the arc defining the trunk, rather than at the origin. The kinematics of this class of robots can therefore be derived through purely geometrical means, without the need of D-H tables and accompanying transformations [6, 7], screw theory [8], or extensive and error-prone computation [9]. Examining Fig. 4, c for a trunk which extends along the +z axis and bends along the direction φ in the xy plane is c = [ rcosφ rsinφ 0] T where r = is the radius of the circle. As shown in the figure, the axis ω about which points on the circle rotate is perpendicular to the circle, computed as R z,90 c then normalizing to yield ω = [ sinφ cosφ 0] T. From [], a rotation R ωθ, about the axis c can be computed by first translating to the origin, performing the rotation, then translating back. Therefore, the desired A is I c Rωθ, 0 I c Rωθ, ( I R, ) c ωθ A = T T T =. T 0 Substituting and recalling = r and noting that the necessary rotation θ about the circle is determined by the ratio of the arc length s of the trunk to the circle s radius r, so that θ = s r, the resulting homogenous transformation matrix is cos φ ( coss ) + sinφcosφ( coss ) sinφ cosφ( coss ) cos φ( coss) + coss A = cosφsins sinφsins 0 0 () cosφsins cosφ( coss) sinφsins sinφ( coss). coss sins 0 Further transformations given in [5] allow computation of the amount of curvature based on the lengths of cables l, and l and radius of the trunk d of l 3 505

The trunk failed to bend in a constant curvature arc because of the low stiffness compared to the heavy end cap causes sag and torsion. Fig.

5 Trunk bending in an arc of constant curvature Heavy end cap causing sag and loading Torsion Gravity r = Fig. 6 Experimental verification with the effect of gravity to verify the validity of the proposed design. The trunk failed to bend in a constant curvature arc because of the low stiffness compared to the heavy end cap causes sag and torsion. Fig. 5 Experimental procedure without the effect of gravity to verify the validity of the proposed design. The large red circle indicates the arc of constant curvature in which trunk is bending. The inverse of the distance between the center and a point on the arc gives the curvature of the trunk. = l + l + l ll l l ll dl ( + l + l) () B. Model verification Under ideal conditions the curvature produced by the trunk should match with the curvature calculated using the formula. An experiment was done where the curvatures of the trunk were measured for various combinations of cable lengths. A paper with circles of different radii drawn on it is used to measure the curvature of the trunk. For a given combination of l, l and l 3, the trunk bends producing a uniform curvature = /r. The shape of the trunk is then matched against the reference circles drawn on the paper as shown in Fig. 5. The curvature of the matching circle is then measured as the curvature of the trunk. The entire experiment is performed by resting the trunk on the ground, therefore eliminating the effect of gravity on the trunk; the frictional effects of the paper are negligible. Table 3 shows calculated and measured for different combinations of the trunk lengths l, l and l 3. As shown in the table, the percentage of error is very small. l in l in TABLE 3 EXPERIMENTAL VERIFICATION l 3 in Error calculated measured in % The same experiment was repeated considering the effect of gravity. This time the trunk fails to bend with a uniform curvature as shown on Fig. 6. The effect of gravity on the trunk is considerable, and the constant curvature assumption does not apply under gravity, because of the low stiffness of the trunk compared to the load carried. The weight of the metal tube fitting at the end of the trunk causes the trunk to deform from its original shape. While this metal fitting can be replaced with a plastic fitting, or the tube can be sealed in some other way without adding additional weight to the trunk, when the trunk is used for practical applications, we expect it to carry a tool at the end of its trunk, which would add weight to the trunk. Though models to estimate the effect of gravity on continuum robots exist [, 3], their complexity is too high to run them in real-time which makes them unsuitable to implement. This motivates the need for development of real-time dynamics for continuum trunks [4]. IV. ELECTRICAL DESIGN This section of the paper presents the design of an electrical system to control a continuum robot. Fig. 7 provides an overview of the electrical setup. A host PC calculates the lengths l, l and l 3 needed to obtain the required shape of a trunk. It then passes these parameters to the PC/04 [5] module, a compact form-factor single board computer suitable for executing real-time applications and supported by a wide variety of off-the-shelf I/O boards. The PC/04 module acts as a driver that actuates the motors to adjust the lengths of cables. The striking feature of this design is the two-level control using a PC and PC/04. A Simulink [6] model is developed on the host PC and converted to executable code using the real Time Workshop [7]. This executable code is then downloaded from the host PC to the PC/04 running the xpc Target real-time kernel. The PC/04 handles the I/O 506

6 Trunk Motor Pressure Regulators H - Bridge Encoders D/A operations through its add-on boards and acts as a driver for the end effectors. The wide variety of commercial, off-theshelf I/O add-on boards for PC/04 systems coupled with the availability of drivers for many of these included in Matlab s xpc Target provides a cost-effective rapid-prototyping environment. The host PC does the major computational work required to calculate the kinematics of a continuum robot and provides a real-time graphical representation of a continuum robot [8]. This graphical model provides essential feedback to the users while they operate the robot. The operator uses a joystick to change the position of the robot. The PC then processes the joystick input and determines the cable lengths to achieve the desired position. Then the PC transmits the necessary control signals to the PC/04 which is connected in a common network via UDP protocol that is widely supported and built in to Matlab s xpc Target. The PC04 has a digital-to-analog converter add-on board that converts the digital control signals to analog voltages that can be used to run electric motors. The analog voltages are given as input to the H-Bridges that drive the motors. The motors are fitted with encoders which are used to measure the individual lengths of cables. The signals from these encoders are read using a quadrature encoder board connected to PC/04. These encoder values are then sent to the host PC. The PC uses this information to calculate the present position of robot in 3D space and updates the graphical model. V. CONCLUSION PC 04 Quad 8 UDP Graphical View Host PC Joystick Fig. 7 Unique two-level electrical design to control a continuum robot. This paper examines two existing mechanical designs and developed a new design combining the simplicity of construction of one with the robustness of the second. This is a low cost design and offers less friction between the cables and cable guides. This is easy to construct and simple to control. To model the behavior of this design a novel derivation of the forward kinematics of a continuum robot was presented. Use of active transformations provides a novel, straightforward derivation for kinematics of continuum robots. Unique experimental examination of the circular arc assumption made by the model reveals that it does not hold in cases where loading due to gravity overcomes the trunk stiffness. This verification procedure is the first approach to verify the constant curvature assumption shared by most of the continuum robots. The method presented in this paper can be applied with readily available laboratory equipment. Finally, this paper presents a unique rapid-prototyping system with which to operate the robot. REFERENCES [] S. Neppalli, B. A. Jones, M. Csencsits, W. McMahan, V. Chitrakaran, M. Grissom, M. Pritts, C. D. Rahn, and I. D. Walker, "OctArm - Soft Robotic Manipulator," in video in Proceedings of the International Conference on Intelligent Robots and Systems San Diego, CA, USA, 007. [] G. Robinson and J. B. C. Davies, "Continuum robots - a state of the art," in Proceedings of the IEEE International Conference on Robotics and Automation, Detroit, Michigan, 999, pp [3] W. McMahan, B. A. Jones, and I. D. Walker, "Design and implementation of a multi-section continuum robot: Air-Octor," in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Edmonton, Canada, 005, pp [4] W. McMahan, B. A. Jones, V. Chitrakaran, M. Csencsits, M. Grissom, M. Pritts, C. D. Rahn, and I. D. Walker, "Field trials and testing of the OctArm continuum manipulator," in Proceedings of the International Conference on Robotics and Automation, Orlando, FL, USA, 006, pp [5] B. A. Jones, W. McMahan, and I. D. Walker, "Design and analysis of a novel pneumatic manipulator," in Proceedings of the 3rd IFAC Symposium on Mechatronic Systems, Sydney, Australia, 004, pp [6] B. A. Jones and I. D. Walker, "Kinematics for Multisection Continuum Robots," IEEE Transactions on Robotics, vol., pp , Feb [7] M. W. Hannan and I. D. Walker, "Kinematics and the Implementation of an elephant's trunk manipulator and other continuum style robots," Journal of Robotic Systems, vol. 0, pp , Feb [8] P. Sears and P. Dupont, "A Steerable Needle Technology Using Curved Concentric s," in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, China, 006, pp [9] Y. Bailly and Y. Amirat, "Modeling and Control of a Hybrid Continuum Active Catheter for Aortic Aneurysm Treatment," in Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain, 005, pp [0] G. Chen, M. T. Pham, and T. Redarce, "Development and kinematic analysis of a silicone-rubber bending tip for colonoscopy," in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, China, 006, pp [] I. A. Gravagne, C. D. Rahn, and I. D. Walker, "Large deflection dynamics and control for planar continuum robots," IEEE/ASME Transactions on Mechatronics, vol. 8, pp , June 003. [] J. M. Selig, "Active versus passive transformations in robotics," IEEE Robotics & Automation Magazine, vol. 3, pp , 006. [3] M. Ivanescu, N. Popescu, and D. Popescu, "A Variable Length Tentacle Manipulator Control System," in Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain, 005, pp [4] E. Tatlicioglu, I. D. Walker, and D. M. Dawson, "Dynamic Modelling for Planar Extensible Continuum Robot Manipulators," in International Conference on Robotics and Automation, Rome, Italy, 007. [5] P. E. Consortium, "PC/04 Specification," November 003. [6] I. The MathWorks, "Simulink reference manual,," Natick, MA, March 005. [7] I. The MathWorks, "The Real-Time workshop user s guide." [8] M. Csencsits, B. A. Jones, and W. McMahan, "User interfaces for continuum robot arms," in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Edmonton, Canada, 005, pp

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